ADAO5 06 AEROSPACE CORP EL SEGUNDO CA ENGINEERING GROUP F/G 21/8.2ONE- AND TWO-PHASE NOZZLE FLOWS.(U)JAN 80 1 CHANG F04701-79C-0080

UNCLASSIFIED TR-OO8O(901-01)-l SD-TR-80-26 NL,'Eh'-IhIhEEIEEIIIIEEEEIIImEEIIIIEEEEEIEEEEEEE//EEIh

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MICROCOPY RESOLUTION TEST CHMRTNAI ON~AL BUJREAU km ST~sARM1%)3-,l

REPORT SD-TR-026

LEVEUV1One-and-Two-Phase Nozzle Flows

I. SHIH CHANG

n Engineering Group"/The Aerospace Corporation

El Segundo, Calif. 90245

-C -C63,• 91 January 1980 JUN l

I81

C

Final Report

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,,,z:, JO 6 -1 1

CJ1 Prepared forSSPACE DIVISION

AIR FORCE SYSTEMS COMMANDm Angeles Air Force Station

P.O. Box 990, Worldway Postal CenterLom Angelee Calif. 90009i is_

"I,

This final report was submitted by The Aerospace Corporation,

El Segundo, CA 90245, under Contract F04701-79-C-0080 with the Space

Division, Deputy for Space Communications Systems, P.O. Box 92960,

Worldway Postal Center, Los Angeles, CA 90009. It was reviewed and

approved for The Aerospace Corporation by E. G. Hertler, Engineering

Group. First Lieutenant J. C. Garcia, YLXT was the Deputy for Tech-

nology project engineer.

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is releasable to the National Technical Information Service (NTIS). At

NTIS, it will be available to the general public, including foreign nations.

This technical report has been reviewed and is approved for publi-

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of the report's findings or conclusions. It is published only for the ex-

change and stimulation of ideas.

/_J3, C. Garcia, i st Lt, USAFProject Engineer Joseph J. Cox, Jr., Lt Col, USAF

Chief, Advanced Technology Division

FOR THE COMMANDER

Burton H. Holaday, Col, USAFDirector of Technology Plansand Analysis

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14. TITLE (and Subtitle) n;_______________ 7FOT I E

tONE- AND TWO-PHASE NOZZLE FLOWS' Ia:w8Jn 80

TR-08(59Ol-#6')- I

6 I-Shih hang

9. PERFORNING ORGANIZATION NAMIE AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

The Aerospace CorporationEl Segundo, Calif. 90245

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j. 14. SUPPLEMENTARY NOTES19. KEY WORDS (Contiuae on reverse old. of noecesay and identify by block .eintber)

Gas-particleTwo- phaseNozzleTransonic FlowCorn utational Method

20. AS Tf ACT (Continue an reverse side it necessary and identify by block ontmbe)

A time-dependent technique, in conjunction with the boundary-fittedcoordinates system, is applied to solve a gas-only one-phase flow and afully-coupled, gas-particle two-phase flow inside nozzles with small throatradii of curvature, steep wall gradients, and submerged configurations. Theemphasis of the study has been placed on one- and two-phase flow in thetransonic region. Various particle sizes and particle mass fractions have been

DFORM 1413UNLSIED4

'1- ..Z. q.&p 6~I 4CURI Y CLASSIFICATION OF THIS PAGEt (e Data tf,'d

UNC LASSIFIEDsCCumiTY CLASSIFICATION4 OF THIS PAOS(Ifbu, Date Entered)

IS. IKY WORDS (Continued)

UTRACT (Continued)

investigated in the two-phase flow. The salient features associated withthe two-phase nozzle flow compared with those of the one-phase flow areillustrated through the calculations for a JPL nozzle configuration, for theTitan III solid rocket motor nozzle, and for the submerged nozzle configu-ration utilized in the Inertial Upper Stage (IUS) solid rocket motor.

UNCLASSIFID* SCURSY CLASSIICAION OFPTRIP PAIIIIMA Doef gEM0ed)

PREFACE

!UThe author is indebted to the late Dr. John Vasiliu for his review of

this report and for his encouragement and helpful suggestions throughout

the study.

t iftif lat to --

- COS5VICO

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A

CONTENTS

PREFACE .......... ..... ... ........ ..... I

I. INTRODUCTION ............. ................ 7

II. GOVERNING EQUATIONS .......... ............ 11

III. NUMERICAL ASPECTS ...... .................. 17

IV. NOZZLE WITH SMALL THROAT RADIUS OFCURVATURE--JPL NOZZLE ......... ............ .23

A. One-Phase Flow ....................... 23B. Two-Phase Flow ............ .............. .26

V. NOZZLE WITH VERY STEEP ENTRANCE--TITAN III MOTOR ............................... .... 41

A. One-Phase Flow ..... ...... ....................... 41

B. Two-Phase Flow ......... ....................... 45

VI. SUBMERGED NOZZLE--IUS SMALL MOTOR .......... ... 51

A. One-Phase Flow ..... ... .................... . . . . 51

B. Two-Phase Flow ....... ....................... ..... 54

VII. CONCLUDING REMARKS ...... ................... ... 61

REFERENCES .......... ............................. ..... 63

-3-

- -

FIGURES

1. Transformation for Boundary-Fitted Coordinate System . . . 18

2. BFC Grid for JPL Nozzle ......... .................... 24

3. Mach Number Distribution at Wall and Centerline forJPL Nozzle (One-Phase Flow) ...... ................. ... 24

4. Mach Number Contour Plot for JPL Nozzle ... .......... .25

5. Mach Number Distribution Throughout the Flow Fieldfor JPL Nozzle ......... ......................... ... 25

6. JPL Nozzle Throat Mach Number at Every IntegrationStep (Two-Phase Flow W./W ff= 30%) .... ............. ... 27

7. JPL Nozzle Mach Number Distribution at Wall andCenterline (Two-Phase Flow W./W = 30%) ..... .......... 28j m

8. Velocity Lag (Two-Phase Flow W./W a 30%) ........... ... 30

9. Gas-to-Particle Temperature Ratio (Two-PhaseFlow W./W = m 30%) ........ ....................... ... 313 m

10. Particle Density Contour for rj = lp and 20/ (Two-PhaseFlow Wj/W m = 30%) ....... ...................... .32

11. Particle Velocity Vector Plot for r f lp and 20# (Two-Phase Flow W./W = 30%) ...... ................ .... 333 m

12. JPL Nozzle Throat Mach Number at Every IntegrationStep (Two-Phase Flow r. .i = ..) .................... ... 353

13. JPL Nozzle Mach Number Distribution at Wall andCenterline (Two-Phase Flow r. 1/1) .............. ...... 363

14. Velocity Lag (Two-Phase Flow r =p) .... ............. ... 38

15. Gas-to-Particle Temperature Ratio (Two-PhaseFlow r. i/..) ........ ........................... .. 39

16. Mach Number Contours for Different Particle Size (Two-Phase Flow W./W = 30%) ...... ................... .... 40

m

-4-

FIGURES (Continued)

17. Mach Number Contours for Different Particle MassFraction (Two-Phase Flow r. = 1AI) .... .............. . 40

18. BFC Grid for Steep Entrance Nozzle ................ ...... 42

19. Mach Number Distribution at Wall and Centerline forSteep Entrance Nozzle ......... ...................... 43

20. Throat Mach Number at Every Integration Step forSteep Entrance Nozzle ........ ..................... 43

21. Mach Number Contour for Steep Entrance Nozzle ........ ... 44

22. Mach Number Pictorial Plot for Steep Entrance Nozzle(One-Phase Flow) ......... ........................ ... 44

23. Mach Number Pictorial Plot for Steep Entrance Nozzle(Two-Phase Flow) .......... ........................ 46

24. Particle Density Contour for Steep Entrance Nozzle ......... 48

25. Particle Density Pictorial Plot for Steep Entrance Nozzle. . 48

z6. Pressure Distribution for Steep Entrance Nozzle ... ....... 49

4 27. Velocity Lag and Temperature Ratio for SteepEntrance Nozzle ....... ......................... ... 49

28. IUS Small Motor Interior Configuration and* Computational Region ...... ...................... .. 52

29. BFC Grid for Small IUS SRM with SubmergedNozzle Block ......... ........................... .... 52

30. Blown-up BFC Grid in the Submerged and ThroatRegion for Small IUS SRM ....... ................... ... 53

31. Throat Mach Number at Every Integration Step forSmall IUS SRM ........ ......................... . 55

32. Mach Number Distribution on the Boundary for SmallIUS SRM Nozzle ......................... 55

-5-

FIGURES (Concluded)

33. Mach Number Contour for Small IUS SRM(One-Phase Flow) ......... ........................ .... 56

34. Velocity Vector Plot in the Submerged and ThroatRegion for Small IUS SRM (One-Phase Flow) ............ ... 56

35. Gas-Phase Velocity Vector Plot in the Submerged andThroat Region for Small IUS SRM (Two-Phase Flow) ......... 58

36. Particle Velocity Vector Plot in the Submerged and ThroatRegion for Small IUS SRM (Two-Phase Flow) .. ......... ... 58

37. Gas-Phase Mach Number Contour for Small IUS SRM(Two-Phase Flow) ........ ........................ .... 59

38. Particle Density Contour for Small IUS SRM(Two-Phase Flow) ........... ........................ 59

39. Velocity Lag and Temperature Ratio for Small IUS SRM . . . 59

pw

'.i . . . ..... :,--6-.

I. INTRODUCTION

The analysis of flow-through rocket motor exhaust nozzles has under-

gone continuous development for many years, since the optional design of

these nozzles is dependent on accurate knowledge of the flow behavior and is

important to the attainment of high thrust efficiencies for launch vehicles.

The classic analytical solution technique based on the series expansion ' 2 has

limited application, as it requires the nozzle entrance to be suitably shaped.

During the past decade the use of computers for the solution of nozzle flow

fields 3 - 8 has been very popular among research engineers, mainly because the

modern high-performance propulsion system, for the sake of length and weight

reduction, usually possesses a nozzle contour with a small throat radius of

curvature, a very steep wall gradient in the entrance region, or a submerged

configuration, and the numerical technique is well-suited for application to

different nozzle geometric configurations. For gas-only one-phase nozzle

flows, various numerical methods used in the past were reviewed in Ref. 9.

1Hopkins, D.E. and D.E. Hill, "Effect of Small Radius of Curvature on Tran.sonic Flow in Axisymmetric Nozzles, " AIAA J., 4(8), Aug. 1966, p. 1337.

2 Kliegel, J. R. and V. Quan, "Convergent-Divergent Nozzle Flows, " AIAA J.,

Sept. 1968, p. 1728.3 Prozan, R.J., reported in "Numerical Solution of the Flowfield in the Throat

Region of a Nozzle," by L.M. Saunders, BSVD-P-66TN-001 (NASA CR82601),Aug. 1966, Brown Engineering Co., Huntsville, Ala.

4 Migdal, D., K. Klein, and G. Moretti, "Time-Dependent Calculations forTransonic Nozzle Flow," AIAA J., 7(l), Feb. 1969, p. 372.

5 Wehofer, S. and W. C. Moger, "Transonic Flow in Conical Convergent andConvergent-Divergent Nozzles with Nonuniform Inlet Conditions, " AIAAPaper No. 70-635.

6 Laval, P., "Time-Dependent Calculation Method for Transonic Nozzle Flows,Lecture Notes in Physics, 8, Jan. 1971, p. 187.

7 Serra, R.A., "Determination of Internal Gas Flows by a Transient NumericalTechnique, " AIAA J., 10(5), May 1972.

8 Cline, M. C., "Computation of Steady Nozzle Flow by a Time-DependentMethod," AIAA J., 12(4), Apr. 1974, p. 419.

9Brown, E. F. and G. L. Hamilton, "A Survey of Methods for Exhaust-NozzleFlow Analysis, " AIAA Paper No. 60, 1975.

-7-

For the solid rocket motor, one of the prime causes of performance

loss and surface damage is the presence of condensed metallic oxide particles

of the combustion products in the flow field. The thermal and velocity lag

associated with the particles often results in decreased nozzle efficiency and

degradation of the motor's effectiveness in converting from thermal to

kinetic energy. Hence, knowledge of the role played by the nongaseous com-

bustion products in the rapid expansion through the throat region and the

qualitative estimation of this influence are essential in the design of a thrust

nozzle. A comprehensive review of investigations involving gas-particle

nozzle flow fields before 1962 is presented in Ref. 10. More recent studies

include the numerical iterative relaxation technique of Ref. 11 and an uncoupledflow model described in Ref. 1Z. While the analysis used in these studies is

helpful in explaining some of the physical processes involved in the gas-

particle flows in the transonic region, both suffer from the same weakness;

i.e., the assumption that the gas-phase streamline coordinates are unaffected

by the presence of particles. This assumption is particularly inappropriate

for a nozzle with a very small throat radius of curvature or very steep wall

gradient, s:.nce the presence of particles can alter the gas flow behavior. The

constant fractional lg and the linear velocity profile assumptions used in

Ref. 13 are not justified a priori. The results obtained or refined from a

1 0 Hoglund, R. F., "Recent Advances in Gas-Particle Nozzle Flows," ARSJournal, May 1962, p. 662.

1 1 Regan, J.F., H.D. Thompson, and R.F. Hoglund, "Two-DimensionalAnalysis of Transonic Gas-Particle Flows in Axisymmetric Nozzles,"J. Spacecraft, 8(4), Apr. 1971, p. 346.

1 2 Jacques, L.J. and J. A.M. Seguin, "Two-Dimensional Transonic Two-PhaseFlow in Axisymmetric Nozzles, " ALAA Paper No. 74-1088, Oct. 1974.

1 3 Kliegel, J. R. and G. R. Nickerson, "Axisymmetric Two-Phase PerfectGas Performance Program, " Report 02874-6006-ROOO, Vols. I and II,Apr. 1967, TRW Systems Group, Redondo Beach, Ca. 90278.

-8-

similar analysis for the transonic regionl 4 are highly uncertain, although

they are the most widely used method in the propulsion industry. The one-

dimensional analysis shown in Refs. 15 and 16, found useful in some areas,

is not applicable to the study of a nozzle with a steep entrance.

In this report, the time-dependent method is applied to the solution of

gas-only one-phase flow and fully coupled gas-particle two-phase flow inside

nozzles of arbitrary geometry. The finite difference scheme and the inlet

boundary conditions incorporated into the flow-field program are shown to

yield good resolution of the entire subsonic-transonic-supersonic flow region.

Moreover, to eliminate the computational difficulty associated with a nozzle

with very steep wall or of a submerged configuration, the Boundary-Fitted-

Coordinates (BFC) system 1 7 is adopted for generating a natural grid. Appli-

cation of the BFC system to the nozzle flow study has greatly enhanced the

capability of the flow-field program to solve problems which hitherto have

not been extensively studied. The emphasis of the study has been placed on

one- and two-phase flow in the transonic region. Various particle sizes and

particle mass fractions have been investigated in the two-phase flow. The

salient features associated with the two-phase nozzle flow compared with

those of the one-phase flow are illustrated through calculations for a JPL

nozzle configuration, for the Titan III solid rocket motor nozzle, and for the

submerged nozzle configuration utilized in the IUS solid rocket motor.

14Coats, D. E., et al., "A Computer Program for the Prediction of SolidPropellant Rocket Motor Performance, " Vols. I, II, and III, AFRPL-TR-75-36. July 1975.

1 5 Soo, S.L., "Gas Dynamic Processes Involving Suspended Solids, " A. I. Ch.E.Journal, 7(3), Sept. 1961, p. 384.

16Hultberg, J.A. and S. L. Soo, "Flow of a Gas-Solid Suspension Through aNozzle, " AIAA Paper No. 65-6, Jan. 1965.

1 7 Thompson, J. F., F. C. Thames, and C. W. Martin, "Boundary-FittedCurvilinear Coordinates Systems for Solution of Partial Differential Equa-tions on Fields Containing Any Number of Arbitrary Two-DimensionalBodies, " NASA CR 2729, July 1977.

-9-

II. GOVERNING EQUATIONS

The usual assumptions are employed below to derive the governing

equations of a gas-particle two-phase flow.

a. Mass conservation is applied to both mixture and individualphases.

b. The mixture flow is adiabatic, i.e., the total energy of the mix-ture is constant.

c. Gas phase is inviscid except for its interaction with the metallizedpart '.les, where the momentum exchange is considered for aviscous gas flow over spherical condensed particles.

d. Energy exchange between the gas and particles occurs throughboth the convective and radiative heat transfer.

e. The particles do not interact with each other, and the collision,

condensation, and decomposition of the particles do not take place.

f. The gas is a perfect gas and is chemically frozen.

g. Volume occupied by the solid particle phase is negligible.

Based on the above assumptions and normalized by the gas-phase stag-

nation state corresponding to the condition at the inlet plane, the governing

equations written in divergence form for an unsteady-state two-phase flow

take the following form:

-t+- +- + = 0 (1)

-11-

rp+

r'5p v rapuv

re r re + (Y1plu

Pj ra'5p iu.(N- 1)

6 62

r'5pj v(N- 1) r P iu v(N- 1)

62r h.(N-1)j~~vvj(-1 6r

G r6P-h.uN-(N-H)

r5Pjujv i(-1 r5pjAj(u-u.i)(N- 1)

r6 (- 1) -r P.A.(v-v.)(N-1) -6-

r6 h iv (N- 1) H - 06jjI N-1

whe re

IN = I one-phase gas-onlyN = 2 two-phase flow

= 0 two-dimension

I I axisymmetryThe nondimensional parameters used here are gas-phase stagnation pressure

Ptl stagnation density ptlI stagnation energy per unit volume el

Ptl/-1), maximum speed V maxl, and stagnation temperature Ttl evaluatedat the inlet plane, so that

= P / Pt = (Y-1)/27 , t = V Ixt/

uu/ ,,u. u./ rr/max J / maxl r

v = V/Vmaxl v. = v./Vmaxl for L reference length scalemx 3(e.g., unit foot)

e = et , h. h/e

where p, u, v,p, and e are the dimensioned density, horizontal x-component

velocity, vertical r-component velocity, pressure, and energy per unit

volume, respectively, for the gas phase; and p., u., vj, and h. are the dimen-

sioned density, x-component velocity, r-component velocity, and energy per

unit volume, respectively, for the particle phase. There are also

Friction term:

j2 - -2 -m jr max!

-13-

w - .

Energy exchange term:

B. =zyrq. Aqj - (T.-T) -g(C.T. - T )43 3 3 3 r3

whe re

g = N UiA/f6 ~jrp g = a Tti/cp Pg9f

qj Aq. u.j(u-u.) + v. (v-v.)

[h./YP. (u.2 + v.)]/W

T T7/Tti=p/p' L)C.fITi p

with

t =dimensioned time

T dimensioned gas temperature

P = gas viscosity

T. =dimensioned particle temperature

m. = particle mass density

r. = particle radius

c p gas specific heat at constant pressure

c. a particle heat capacity

Y = gas specific heat ratio

0= Stefan- Boltzmann constant

e. a particle emissivity

C a gas emissivity

-14-

r = dimensioned radial coordinate

Pr a gas Prandtl number

x z dimensioned axial coordinate

The momentum transfer parameter f. is defined as

j D DStoke s

where C D is the drag coefficient based on C. B. Henderson's correlation equa-

tion 1 8 for spheres in continuum and rarefied flows, and CDStokes a Re./24 is

the Stokes law of drag coefficient for spheres in creeping motion.

The heat transfer parameter, particle Nusselt number, is taken as

0.55 0.33N .i 2 + 0. 459 R 1j0.3r

The particle Reynolds number is based on the relative speed 4qj . =

Aj a-Vu-) + (v-v.)2 and the particle radius r and is definedIj I Maxl 1 d sdeie

as follows:

Rej 2 AqfPf g max I

The gas viscosity is evaluated from

tiI P

where ut, is the gas viscosity at the stagnation temperature Tt1 corresponding

to the inlet condition, and A is an input constant.

1 8 Henderson, C. B., "Drag Coefficients of Spheres in Continuum and RarefiedFlows, " AIAA J., 14(6), June 1976, p. 707.

-15-

It is often debated which form of the equation the drag coefficient CD

and the particle Nusselt number Nu. should take. The correlating equations

of Ref. 18 provide accurate representations of sphere drag coefficients

over a wide range of flow conditions. The simple form of the particle Nusselt

number is adopted from Ref. 19. If more advanced parametric equations are

available, they can be incorporated easily into the present analysis without

much modification.

The computer program Dusty Transonic Internal Flows (DTIF) has been

developed so that, for the gas-only one-phase flow (N=I), all the particle

phase calculations are bypassed.

19Carlson, D.J. and R.F. Hoglund, "Particle Drag and Heat Transfer inRocket Nozzles, " AIAA J., 2(11), 1964, p. 1980.

-16-

wool., .

III. NUMERICAL ASPECTS

From a general arbitrary nozzle configuration in the physical plane x, r

the transformation to a grid with uniform square mesh in the computational

plane , can be accomplished by using BFC; this requires the solution of

two elliptical partial differential equations with Dirichlet boundary conditions.17

Figure 1 illustrates the transformation relationship. The solution utilizing

the successive over-relaxation (SOR) method for generating the boundary-

fitted coordinates is carried out by the TOMCAT program, and the scale17

factors for transformation are computed in the FATCAT program.

Formally applying the chain rule of change of independent variables for

Eq. (1) results in the following conservation laws in the Cf plane:

)E F G+ +-T + -q + H 0 (2)

whe re

E --EJ2

F F r -_xfG F r. + Gx4.

H =HJ2

and JZ = xr - yris the Jacobian of transformation. For a particular

nozzle geometry and transformation, the Jacobian and the partial derivatives

are computed in the FATCAT'" program and stored on disk for flow field study.

For a simple region such as the nozzle geometry considered herein, theFATCAT program gives the values of scale factors at the corner pointstwice as large as they should be. This has been corrected for the nozzleapplication in this study.

-17-

COMPUTATIONAL PLANE lt)

Fig. 1. Transformation for Boundary-FittedCoordinate System

Through insertion of the unsteady term in the governing Eq. (1), the

differential equation is cast into hyperbolic type; therefore, the complication

associated with the mixed flow phenomenon necessarily existing in a steady-

state analysis is eliminated. The MacCormack finite difference scheme

2 0

which has been applied successfully to nozzle flow problems2 1 is adopted here

for the solution of the partial differential Eq. (2).

For one-phase flow, the initial condition is based on a one-dimensional

isentropic analysis with the flow vector set to the local inclination angle from

linear interpolation between the lower and upper wall slopes along the same

grid line (constant ). The converged one-phase results serve, then, as the

initial guess for the gas-phase data in the two-phase flow.

For the particle-phase arrays of initial velocity lags X and temperature

ratios AT are chosen, and the initial condition (guess) is

p. - p u. = UX v v=

T. = T/X h. = Yp.[WT. + (u. + v 2

where O = W./W is the particle mass fraction and WJ = -C-. is the ratio of3 m i pparticle heat capacity to gas specific heat at constant pressure.

A unit velocity lag and temperature ratio as an initial guess of particle

phase are satisfactory for this study.

The initial guess based on the so-called Equilibrium Gas-Particle

Mixture (EGPM), which is itself a physically meaningless term with its isen-

tropic index derived from

Ym I +,,,OU -0)

is not appropriate for the fully coupled two-phase flow study.

20MacCormack, R.W., "The Effect of Viscosity in Hypervelocity Impact

Cratering, " AIAA Paper 69-354, May 1969.21Chang, I-Shih, "Three-Dimensional Supersonic Internal Flows," AIAA

Paper 76-423, July 1976.

-19-

IA&

The exit boundary condition is based on a simple linear extrapolation

since the mixture flow is assumed to be supersonic at the exit plane, and the

error generated from the extrapolation is not expected to propagate back

and affect the upstream results.

The inlet boundary condition for horizontal inflow is computed from a

characteristics formulation similar to that of Ref. 7, which provides fairly

smooth subsonic flow in the physical domain. For radial inflow, e.g., the

IUS motor studied herein, the experimentally evaluated propellant burning

rate and chamber pressure/temperature data supply needed information at

the propellant burning surface, and a linear interpolation for smoothing flow

variables at the grid line adjacent to the propellant burning surface is required

to avoid instability in the inlet region.

For a nozzle with a centerbody, the flow variables at the boundary are

obtained from linear extrapolation of the data from two adjacent interior

points and then modified by the local tangency condition. Without the center-

body in the axisymmetric nozzle, the flow variables take undetermined form

at the centerline. The standard L'Hospital's rule with the symmetry con-

sideration is used to evaluate all the physical flow variables except the radial

velocity, which is zero at the singular centerline. Also, if notice is taken

that all the conservative variables at the centerline are zero except H 3 -P

(i. e., no restriction is imposed on the gas pressure change at the centerline),

a smoothing process involving a linear interpolation for resetting the gas-

phase radial velocity v at the first interior point above the centerline is

found helpful in stabilizing the solution.

It is important also to retain the fourth-order damping terms to the

second-order MacCormack finite difference scheme for the unsteady state

application in order to eliminate the nonlinear instability. The formulation

adopted here is similar to that of Ref. 22 with a damping coefficient equal to

0.01.2 2 Kutler, P., L. Sakell, and G. Aiello, "On the Shock-on-Shock Interaction

Problem," AIAA Paper No. 74-524, June 1974.

-20-

The integration step size is governed by the local CFL condition

similar to that used in Ref. 5 and determined by the following expressions:

At = min

q +a

whe re

AL = j/ +r)Z , q = /UZ 1 +VI

and a f VT(Y-l)/2 is the dimensionless local sonic speed.

-21-

IV. NOZZLE WITH SMALL THROAT RADIUS OFCURVATURE--JPL NOZZLE

A. ONE-PHASE FLOWZ3

The compressible flow inside the JPL axisymmetric nozzle with

450 entrance and 150 exit straight wall tangent to a circular throat (with ratio

of throat radius of curvature to throat height = 0. 625) provides a classic

comparison for the present nozzle flow study. Figure 2 shows the physical

grid generated from the boundary-fitted coordinates system mentioned above.

The computed Mach number distributions along the wall and along the

centerline are illustrated in Fig. 3; the test data are also shown for compari-

son. Figure 4 is the Mach number contour plot, and Fig. 5 shows the Mach

number at all the grid points in the physical plane. The theoretical gas-only

one-phase result from this study agrees very well with the test data in the

entire subsonic-transonic-supersonic flow region. This can be attributed to

the good resolution of the boundary flow variables through the use of

boundary-aligned grid arrangement. Note the smoothness of the Mach num-

ber distribution in the subsonic region computed by the present method.

The recompression waves in the supersonic region, which necessarily occur

due to over-expansion of the flow near the wall downstream of the throat with

small radius of curvature, eventually coalesce into a shock wave near the

centerline in the far-downstream region. This flow behavior has been ob-

served in the test 7 and is visible from Figs. 4 and 5.

The convergence criterion used in all the calculations shown herein

requires that the difference in Mach number is at least 0. 0 1% and in the mass

flow rate is 0. 001% at the throat for three consecutive time integration steps.

For the JPL nozzle with 61 x 31 grid points, the converged solution requires

623 integration steps and takes 6 min, 17 sec execution time on the CDC 7600

computer. The theoretical results agree well with the test data for this23 Cuffel, R. F., L. H. Back, and P. F. Massier, "Transonic Flowfield in aSupersonic Nozzle with Small Throat Radius of Curvature," AIAA J., 7(7),

July 1969, p. 1364.

-2

-_7n

2.5

450

0.0-3.0 iin.) 1.5

Fig. 2. BFC Grid for JPL Nozzle

2-4

2.0- THEORETICAL RESULTS WALL

_ A WALL REF 231.6 e CENTER TEST DATA

LINE1.2- A WALL REF.7

= 0 CENTER- TEST DATA

0.8 _ LINE CENTER _LINE1.

0.4 -

-3.0 -2.0 -1.0 0.0 1.0

" (in.I

Fig. 3. Mach Number Distribution at Wall and Centerline forJPL Nozzle (One-Phase Flow)

-24-

2.5

CONTOUR LINE INCREMENIAM0.1

Y=1.4

0.0-3.0 (1in.) M 1.0 1.5

Fig. 4. Mach Number Contour Plot for JPL Nozzle

MACHNo. = 2.412

MACHNo.= 0.01.5

2.5(in.)

(in.)0.0 -3.0

Fig. 5. Mach Number Distribution Throughout theFlow Field for JPL Nozzle

-25-

nozzle study, thus assuring further application of the computer code to

other nozzle configurations.

B. TWO-PHASE FLOW

The same flow field program is applied to the fully-coupled gas-particle

two-phase nozzle flow with the two-phase index N in Eq. (1) set to 2. For

the two-phase flow calculation, the following data are adopted.

Gas Phase (Air) Particle Phase

= 0.255 Btu/lb - 0 R = 0.33 Btu/lb -ORp m m

pt, = 1. 8x10 - 5 lb /ft-sec m. = 250 lb /ft 3

= 1.4, Pr = 0.74

A = 0.6

The chamber condition is taken to be T = 1000 R, P = 150 psia.

Also, f. = 0. 1 and E = 0.05 are used for the radiative heat exchange betweenjthe gas and spherical particles.

The previously computed one-phase flow result is taken as the initial

guess for the two-phase flow. Different particle sizes and particle mass

fractions W./W are calculated. Figure 6 shows the variation of the com-j mputed throat Mach number along the wall and along the centerline at each

timewise integration step (iteration) for various particle sizes at the same

W.IW = 30%. At 300 iterations the two-phase transonic flow region isSmessentially established. Further integration does not produce appreciable

change in the throat flow field, as is evidenced from the continued calcula-

tion with r. = 1/i to 600 integration steps. Figure 7 shows the computed wallJand centerline gas phase Mach number distribution for various particle sizes

with the same Wj/Wm = 30%. For comparison, the previously computed

-26-

1.1

0.20

1.3

W 1/m=

0-.7

2.4

2.0- WALL ONE-PHASE

1.6- W. 20p±- W I -30%

~1.2 m

0.8->. lp =T

2.4

CENTERLINE/2.0- ONE-PHASE//

W.

m / 20p

0.8- y1.4 lo

0.4- p=

0.0 _-3 -2 -1 0 1

x lin.)

Fig. 7. JPL Nozzle Mach Number Distribution at Wall andCenterline (Two-Phase Flow W.i/W = 30%)

-28-

gas-only one-phase results are plotted as dashed curves. Some of the

features associated with the fully-coupled two-phase nozzle flow in compari-

son with that of gas-only one-phase flow are found in Figs. 6 and 7. While

lower gas speed is observed both on the wall and centerline in the two-phase

flow field than that in the gas-only one-phase, the small-sized particles act

more effectively to slow down the gas-phase expansion than that of large-

sized particles for the same particle mass fraction. This is physically

correct, since for the same particle mass fraction the total particle surface

area effective for momentum and energy exchange between gas and particles

is greater in a two-phase flow field involving smaller diameter particles.

Figure 8 gives the velocity lag qj/q at the throat and at the exit plane,

respectively, and Fig. 9 shows the results for the temperature ratio T/Tj.The temperature ratio defined here illustrates the thermal lag between the

particle and gas and is less sensitive to the small variation in the computed

numerics than the commonly used thermal lag (1-T)/(I-T), especially in

the low subsonic region when both gas and particle temperatures are very

close to the reference gas stagnation temperature at the inlet plane. The

large particle in the two-phase flow field lags much more behind the gas

phase flow and exhibits a wider region of particle free zone than that of the

small particle. Moreover, in comparison with the present fully coupled

two-phase analysis, the constant fractional lag assumption used in Ref. 13

is justified only for a two-phase flow with high loading ratio of very small

particles.

The effect of different particle sizes in the two-phase flow can be

visualized best by comparison of the calculated particle density contour and

particle-phase velocity vector plot for the small (r. 1I) and the large

(r. z 201J) particles depicted in Figs. 10 and II. These figures show that for

the flow with lp radius particles a sharp change in particle density is obtained

near the upper wall downstream of the throat, and the particle density

drastically decreases to a small value. But it does not vanish exactly at the

wall. With the large (r. - 2011) particle flow, however, a very distinctive

-29-

0.9 ,

0.1

0.50U 0.2 0.4 0.6 0.8

1.07EXIT PLANE t

0.9

0.7 20

0.60.0 0.2 0.4 0.6 0.8 1.0 1.2

r in.)

Fig. 8. Velocity Lag (Two-Phase Flow W./W = 30%)

-30-

po

1.0 TRA

S0.9 5p

0.8L0.0 0.2 0.4 0.6 0.8

0.60

0.6

0.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Fig. 9. Gas-to-Particle Temperature Ratio(Two-Phase Flow W./ W = 30%)

-31-

2.5

r=

Ap. 0.04

P .=0.4

2.5

-*03.0 flin.J zj . 4 1.5

Fig. 10. Particle Density Contour for r. =l1 and 20/1(Two-Phase Flow WjI/Wm = 30A%)

-32-

03.0 x(n.) 1.5

r.=20 F

$ .. A:PARTICL[

-3.0 ~1n)1.5

Fig. 11. Particle Velocity Vector Plot for r. 1p/. and 20)1(Two-Phase Flow W.j/W m 30%) J

-33-

particle-free zone appears in the calculated result and can be seen from

either Fig. 10 or Fig. 11. Since the heavier particles apparently cannot

effectively turn around the throat corner with small throat radius of curva-

ture and evidently tend to cluster near the centerline, there are essentially

no particles present near the wall downstream of the throat to slow down the

gas expansion. This explains why, in Fig. 7, the difference between the

Mach numbers in one-phase and in large particle two-phase flows near the

nozzle lip region is virtually nil, but not so at the exit centerline region.

The variation of the computed throat gas phase Mach number along the

wall and along the centerline at each timewise integration step for a different

particle mass fraction W./W at a given particle radius r. = I/1 is indicatedJ m j

in Fig. 12. Figure 13 shows the corresponding wall and centerline gas phase

Mach number distribution. As before, the dashed curves are the results for

the gas-only one-phase flow. It is obvious from these figures that a reduc-

tion of particle mass fraction immediately reduces the two-phase "drag"

effect. A high particle mass fraction (Wj/W m _ 45%) produces an entirely

subsonic flow at the geometric throat for the nozzle geometry considered.

It is then possible to adjust the exit Mach number from supersonic all the

way down to subsonic by varying the particle mass fraction and/or the particle

size. The lip shock extending to the exhaust plume field which occurs in the24,25

over- and under-expanded case ' could also be weakened or eliminated

through the particle drag effect.

A close look at Figs. 6 and 12 also reveals the fact that care must betaken in the case of large particle mass fraction flows (W./W > 45%) and

3 mvery small-sized particle flows (r. < 0. 511) in starting from the one-phase

initial guess. A sharp drop in the wall Mach number at or near downstream

2 4 Chow, W. L. and I-Shih Chang, "Mach Reflection Associated with Over-

expanded Nozzle Free Jet Flows, " AIAA J., 13(6), June 1975.2 5 Chang, I-Shih and W. L. Chow, "Mach Disc from Underexpanded Axi-

symmetric Nozzle Flow, " AIAA J., 12, Aug. 1974, p. 1079.

-34-

1.4

1.3 1%

1.21%

1.11

~1.0 W

0.9 y 141

0.8 CNELN 0

0.7

0.6 -LC0 100 200 300 600

INTEGRATION STEPS

Fig. 12. JPL Nozzle Throat Mach Number at EveryIntegration Step (Two-Phase Flow r.j = I#L)

-35-

2.4 1

2.0 ONE-PHASEWALL

1.6 T= 1p1

M =.21.4 15%~1.2 30%

0.8 V 5=WW

0.4

0.0

2.4

2.0 CENTERLINEONE-PHASE

1.6 1 .

* - 1.2

5%

0.8 30%

0.445=W

0.01-3 -2 -1 0 1

(I n.)

Fig. 13. .1 Pl, Nozzle Mach Number Distribution at wall andlCenterline (Two-Phase Flow r. 1I)

-36 -

of the throat can be expected, especially when the throat radius of curvature

is small, necessitating the use of small time marching step at the beginning

of the two-phase calculation. Furthermore, the possibility of subsonic flow

occurring at the exit plane requires modification of the supersonic down-

stream boundary condition or the extention of the exit boundary to a farther

downstream location.

Figures 14 and 15 show the computed velocity lag and temperature

ratio for various particle mass fractions at a fixed r. = 1/1. Figure 16 sum-

marizes several Mach number contours, M = 0. 1, 0.2, 0. 5, 1.0, and 1.5,for various particle sizes at a fixed ratio Wj/W - 30%, and Fig. 17 is the

Smsimilar result for various particle mass flow ratios at a fixed radius r. -1/.

As before, the dashed curves are the results for a gas-only one-phase flow.

A typical two-phase run involving 300 integration steps takes approximately

7 min execution time on the CDC 7600 computer.

-37-

0.90

0.0 0 % 04 0. .

0.0 0.2 0.4 0.6 0.8 1. 12

Fi1.0 Ve1ct La 1ToPhs Flo 1 1/1EXIT LANE 45%

0.95-38-

1.004

0.95

0.0 0. 0.4 0.6 0.

W -M

0.85 I

0.0 0.2 0.4 0.6 0.8 1. 1.Thin.)

Fi.915. a-oPril prtr ai

(XTwoPAE Fo .110.93

-39-%

00,M 0.1 0.? 0.5 1.0 1.5

Fig. 16. Mach Number Contours for Different Particle Size(Two-Phase Flow W./W 300)

mJ

25 ONE __I PHASE W

30%_\0

45%

M =0.1 0.2 0.5 1.0 1.5

Fig. 17. Mach Number Contours for Different Particle MassFraction (Two-Phase Flow rV. =3 )

_40-

V. NOZZLE WITH VERY STEEP ENTRANCE--TITAN III MOTOR

A. ONE-PHASE FLOW

A severe test of the present numerical technique is the solution of the

compressible flow field inside a nozzle with near vertical entrance region

shown in Fig. 18, which also illustrates the grid generated from the boundary-

fitted coordinates system. The nozzle steep entrance contour is that of the26

Titan III solid rocket motor nozzle. The specific heat for the combustion

products is Y = 1. 19. The computed wall and centerline Mach number distri-

bution is given in Fig. 19. The same nozzle was analyzed with Y= 1.4, and

the results are also plotted on the same figure which serves to illustrate the

effect of different y on the Mach number distribution. Figure 20 depicts the

computed throat Mach number at every integration step. The converged

one-phase flow solution for Y = 1. 19 requires 890 integration steps and takes

8 min, 58 sec on the CDC 7600 computer. Figure 21 is the Mach number

contour plot, and Fig. Z2 shows the Mach number at all the grid points. The

sonic point on the wall has been observed to be far upstream of the throat,

which indicates that higher heating rate occurs farther upstream of the throat

than that expected from the simple one-dimensional analysis. This partially

explains why, in past full-scale firings, the Titan III motor nozzle was

ablated much more in a region far upstream of the throat than at the throat.

Cold-flow tests were recently conducted at the Chemical Systems Division

of United Technology Corporation for the 60 canted Titan III solid rocket

motor using nitrogen with Y- 1. 4; the wall Mach number at the throat was

measured 2 7 to be 1. 52 which agrees fairly well with the computed value 1.6

for the axisymmetric nozzle.

P6Private communication with Chemical Systems Division of United Tech-nology Corporation on Titan III Engineering Drawings, Nov. 1978.

2 7 Private communication with R. Dunlap, Chemical Systems Division ofUnited Technology Corporation, Nov. 1978.

-41-

58.991

0.000-53.1626 ~(n)30.0

Fig. 18. BEC Grid for Steep Entrance Nozzle

-42-

2.0

TWO-PHASE X 1.19--- ONE-PHASE Y= 1.19

1.5 ONE-PHASE Y = 1.4 oil /7

WALL- /1.0 -

CENTER0.5 - LINE

- ~ ~ JTHROAT0.-60.0 -40.0 -20.0 0.0 20.0

X(in.I

Fig. 19. Mach Number Distribution at Wall and Centerline forSteep Entrance Nozzle

1.6 1

1.4- WALLI

1.2 ONE-PHASE TWO-PHASEM Y= 1.19)I

1.0,-CENTERLINE

0.80.6 I"

U 200 400 600 800 100 300 500INTEGRATION STEPS

Fig. 20. Throat Mach Number at Every Integration Step forSteep Entrance Nozzle

-43-

58.99158.991 TWO-PHASE

ONE-PHASE(Y= 1.19)

j A M =0.2 !

0.0 i

-53.1626 M 1.0 30.0V'in.)

Fig. 21. Mach Number Contour for Steep Entrance Nozzle

MACH No.= 1.9219

= 1.19

58.99158.91 MACH No.= 0.0000

30.000

MACH -53.1626

Fig. ZZ. Mach Number Pictorial Plot for Steep Entrance Nozzle(One-Phase Flow)

-44-

Note that no converged solution is possible with the conventional grid

with vertical axial stations, such as that used in Refs. 4 through 8 and in

most of the nozzle studies reported thus far, for the nozzle with a steep wall

slope like that of the Titan III motor. It is the author's experience that when

the nozzle wall slope is greater than z60 ° , the conventional grid with vertical

axial station cannot handle the drastic change in flow properties along the

steep wall, and this results in numerical instability. On the contrary, no

difficulty is encountered in the calculation with the grid generated from the

boundary-fitted coordinates system.

B. TWO-PHASE FLOW

The two-phase flow data for the Titan III motor are as follows:

Gas Phase Particle Phase

C = 0.64 Btu/lb -0 R C. = 0.33 Btu/lb - 0 Rp m m

jt = 5.97x0 - lb m/ft-sec m. = 200 lb /ft3

m m

P = 0.45 r = 6jur

A = 0.664, Y = 1.19 W./W = 28.8%m

The chamber condition is Tt1 a 5890 0 R, Pt fi 600 psia.

The variation of the computed throat gas-phase Mach number along the

wall and at the centerline at every integration step based on the initial guess

from the previously computed one-phase results is also illustrated in Fig. 20.

At the end of 500 integration steps the gas-phase Mach number distribution is

shown in Fig. 19 for comparison with that of one-phase flow. The gas-phase

Mach number contour is plotted in Fig. 21 and the corresponding gas-phase

Mach number distribution throughout the flow field is depicted in Fig. 23.

-45-

MACH NO.= 1.5523

58.991 MACH No. = 0.0000

30.0

0.0 -53.1626

Fig. 23. Mach Number Pictorial Plot for SteepEntrance Nozzle (Two-Phase Flow)

-46-

woo-'

The particle density contour and distribution are plotted in Figs. 24 and 25.

The gas-phase pressure field has not been altered much by the introduction

of particles in the flow field, and the pressure distribution is shown in

Fig. 26. The velocity and temperature ratio are indicated in Fig. 27. The

calculation of 500 integration steps for the two-phase flow takes 10 min,

37 sec execution time on the CDC 7600 computer.

Particles are likely to impinge on the steep wall in the entrance region.

The present analysis does not incorporate any pertinent particle impinge-

ment model for calculating the erosion caused by such impingement. Never-

theless, the particle density contour obtained from this study does show

regions of high particle concentration which may affect the results from

boundary layer calculations and thereby the results of transient in-depth

thermal analyses for the prediction of nozzle wall temperature.

-47-

-5--991

0.0.4AP 00

-53.1626 x fin.) Z 0.2 3.

Fig. 24. Particle Density Contour for Steep Entrance Nozzle

RHOP/ RHO0 = 0.102458.991RHOPIRHOO = 0.0000

30.0

0.0 -53.1626

Fig. 25. Particle Density Pictorial Plot forSteep Entrance Nozzle

-48-

1.0l = --" '= .

0.90.9 -- TWO-PHASE CENTER-

0ONE-PHASE LINE

0.7 - Y= 1.19)

0.6-

0.5 - WALL

0.4-

0.3-

0.2 - THROAT

0.1 I I I I I-60 -40 -20 0 20

i Iin.)

Fig. 26. Pressure Distribution for Steep Entrance Nozzle

THROAT-- - EXIT PLANE

1.00

~0.95 T ---

S*0.90-_0-.8511 1 1 1 f T

0 4 8 12 16 20 24TIM.)

Fig. 27. Velocity Lag and Temperature Ratio forSteep Entrance Nozzle

-49-

VI. SUBMERGED NOZZLE--IUS SMALL MOTOR

A. ONE-PHASE FLOW

It has long been recognized that the solution of the flow field inside

rocket motors with a submerged nozzle configuration constitutes an impor-

tant phase of the flow-field study. Both large and small IUS solid rocket

motors 2 8 have a submerged nozzle. The IUS will be used as an upper stage

for both the Titan III and the Shuttle. In the past, due to the difficulty of

analyzing the internal flow field for IUS-like motors with complicated geo-

metry, various approximations have been adopted, and the accuracy of the

pressure and heat transfer predictions on the submerged nozzle surface was

therefore uncertain. Although the viscous effect would probably dominate

some part of the submerged flow region in the gas-only one-phase flow, the

inviscid flow solution shown here constitutes a first attempt toward a

complete viscous flow solution in future studies.

The IUS small motor interior geometry including the igniter, submerged

nozzle block, and a propellant burning surface is illustrated in Fig. 28, where

the physical region for computation has been identified by heavy solid lines.

The grid generated from the boundary-fitted coordinates system is depicted

in Fig. 29. The flow region is bounded by: (a) the motor case, (b) the

motor centerline of symmetry, (c) the igniter surface, (d) the supersonic

exit plane, and (e) the propellant surface with radial mass inflow. The region

depicted in Fig. 29 incorporates the entire subsonic flow region without

introducing a fictitious vertical inlet boundary. The blownup figure for the

submerged and throat region is shown in Fig. 30.

The propellant burning rate for the small IUS solid rocket motor is

found to be 0.206 in/sec at Ttl = 5985 0 R and P = 410 psia, which results in

2 8 "Inertial Upper Stage SRM-II Baseline Design Review, " Chemical Systems

Division of United Technology Corporation, Dec. 1978.

-51-

74.22 in.

Fig.~~~~~~~ U 28.L USmalMOTOR Itro ofgrto n

AEXI

Fig. 2. BFC Gridl foor Smaio CUonfM igrtho nSomutaterged Noeglock

x -344in. =-52-

Fig. 30. Blown-up BFC Grid in the Submerged andThroat Region for Small IUS SRM

-53-

the fixed inlet condition at the propellant burning surface v = 11.53 ft/sec

(M = 0.00322) for C- --" 0.45 Btu/lb -OR and Y = 1. 19. The Mach numberp min the junction region, where the igniter joins the zero radius centerline,

was computed incorrectly in an earlier study. In this report, the boundary

points with the radial coordinate smaller than the radial length of the adjacent

finite difference mesh are treated as the centerline points of zero radius, and

L'Hospital's rule is conveniently applied thereby avoiding numerical error

resulting from decoding conservative variables divided by a very small

number (small radial coordinate).

Unlike the previous two nozzle flows, a stricter convergence criterion

is deemed necessary for the submerged nozzle calculation and requires that

the difference in Mach number be less than 0. 001% and in mass flow rate

less than 0. 001% at the throat for three consecutive integration steps. For

the 61 x 31 grid points shown in Fig. 29, the converged solution requires

4487 integration steps and takes 38 min, 16 sec execution time on the CDC

7600 computer.

Figure 31 indicates the computed throat Mach number at every inte-

gration step. The IUS solid rocket motor has a throat with a large radius of

curvature; the computed Mach number at the throat is 1.071 on the wall

and 0. 947 at the centerline which are close to a uniform one-dimensional

flow. Figure 32 depicts the Mach number distribution along the boundary,

while the computed Mach number contour is plotted in Fig. 33. Figure 34

shows the velocity vector plot in the submerged and throat region.

B. TWO-PHASE FLOW

The following data are used for the two-phase flow inside the small

IUS motor:

-54-

1.3

1.0 WL

0.9

0.8

0.7

00 2000 400I0 200 400 60i0 80~0 1000

INTEGRATION STEPS

Fig. 31. Throat Mach Number at Every Integration Step forSmall IUS SRM WL

2.2 T~WO-PHASE WL

1.8 ONE-PHASE

ZE 1.4CENTERLINE

1.0 IGNITER TIP0.6 REGION

/ %,NCENERLINE THROAT-0.2 1\f

-35 -30 -25 -20 -15 -10 -5 0 5V (in.)

Fig. 32. Mach Number Distribution on the Boundary forSmall IUS SRM Nozzle

-55-

M 0.1M 0.2 A\M0.1M=.1 M=1

Fig. 33. Mach Number Contour for Small IUS SRM(One-Phase Flow)

F. 34. Veoct VetrPo nteSbegdadTra

Reio for Smal IUS S OePhs lw

-56.

b,. 4

Gas Phase Particle Phase

C 0.45 Btu/lb -0R C. = 0.32 Btu/lb -'Rp m m

=t ; 5.674x10-5 lb /ft-sec m. = 200 lb Ift3

ti m m

P = 0.269 W./W = 30%r m

A = 0.65 r. 2 2.5#

Y = 1.19

The chamber condition is Ttl - 5985 0 R, PZj = 410 psia.

The throat Mach number at every integration step is shown in Fig. 31

for easy comparison with the one-phase solution. The Mach number distri-

bution along the boundary surface is indicated in Fig. 32. The gas-phase

Mach number at exit plane is 1. 57 at centerline and 2.41 at wall for the two-

phase flow; these are smaller than the corresponding Mach numbers of

2.25 and 2.57 found in the gas-only one-phase flow, due to the presence of

solid particles in the flow field. This implies that an IUS solid rocket motor

nozzle flow field and heat transfer analysis based solely on a gas-only one-

phase study will be in error. Figure 35 shows the gas-phase and Fig. 36 the

particle velocity vector plot. A distinctive particle-free zone is visible from

the particle velocity vector plot of Fig. 36. Figure 37 is the computed gas-

phase Mach number contour, and Fig. 38 is the particle density contour in

the two-phase flow. The velocity lag and gas-to-particle temperature ratio at

throat (where r = 2. 15 in.) and exit plane (where r = 4.05 in.) are shown int eFig. 39.

Although the submerged nozzle configuration is complex and the govern-

ing two-phase partial differential equations are highly nonlinear, no compu-

tational difficulty has been encountered during the course of this study, and the

-57-

-------------------------------------------------.r

Fig. 3 5. Gas-Phase Velocity Vector Plot in the Submerged andThroat Region for Small IUS SRM (Two-Phase Flow)

44-

* V Z

Vig 36. PaticeVlctyVco lo nteSumren

Thoa VeinfrSal U Tw-hs lw

V ~ -58-

M= 1.0

Fig. 37. Gas-Phase Mach Number Contour for SmallIUS SRM (Two-Phase Flow)

APi= 0.04

Fig. 38. Particle Density Contour for Small IUS SRM(Two-Phase Flow)

1.0 -

TTHROAT EXIT0.9 , PLANE

Itqj

0.8 I q4 I I i L0 1 2 3 4

Fig. 39. Velocity Lag and Temperature Ratio forSmall IUS SRM

-59-

timewise integration has been carried out in a straightfoward manner.

For the submerged small IUS solid rocket motor nozzle, the two-phase

flow field calculation of 1000 integration steps takes 23 min, 2 sec execution

time on the CDC 7600 computer. All the two-phase flow features mentioned

previously for the JPL and Titan III nozzle are equally applicable to this

submerged IUS solid rocket motor nozzle.

-60-

VII. CONCLUDING REMARKS

The following conclusions have been reached as a result of this study:

a. A time-dependent technique with the MacCormack finite differencescheme provides stable integration for both one- and two-phasenozzle flow equations.

b. Thu utilization of the BFC system enhances the capability of theprogram to the solution of flow inside nozzles with complexgeometry.

c. Imbedded shock can occur in the region downstream of the nozzlegeometric throat for the flow inside nozzle with small throatradius of curvature.

d. The small-sized particles act more effectively to slow down thegas-phase expansion than that of large-sized particles for thesame particle mass fraction.

e. For a two-phase flow with high particle loading ratio, the gas-phase can become subsonic at the geometric throat.

f. The computed one- and two-phase results are important fornozzle wall heat transfer and ablation study.

g. In general, the assumption of constant fractional lag is not justi-fied for a two-phase transonic flow. The prediction of the gas-particle flow field requires that the proper momentum and energyexchange between the gas and particles, such as the fully coupledsolution presented in this study, be taken into account.

-61 -

REFERENCES

1. Hopkins, D.E. and D.E. Hill, "Effect of Small Radius of Curvatureon Transonic Flow in Axisymmetric Nozzles, " AIAA J., 4(8), Aug.1966, p. 1337.

2. Kliegel, J.R. and V. Quan, "Convergent-Divergent Nozzle Flows,AIAA J., Sept. 1968, p. 1728.

3. Prozan, R.J., reported in "Numerical Solution of the Flowfield inthe Throat Region of a Nozzle," by L.M. Saunders, BSVD-P-66TN-001(NASA CR82601), Aug. 1966, Brown Engineering Co., Huntsville, Ala.

4. Migdal, D., K. Klein, and G. Moretti, "Time-Dependent Calculationsfor Transonic Nozzle Flow," AIAA J., 7(1), Feb. 1969, p. 372.

5. Wehofer, S. and W.C. Moger, "Transonic Flow in Conical Convergentand Convergent-Divergent Nozzles with Nonuniform Inlet Conditions, "AIAA Paper No. 70-635.

6. Laval, P., "Time-Dependent Calculation Method for Transonic NozzleFlows, " Lecture Notes in Physics, 8, Jan. 1971, p. 187.

7. Serra, R. A., "Determination of Internal Gas Flows by a TransientNumerical Technique," AIAA J., 10(5), May 1972.

8. Cline, M. C., "Computation of Steady Nozzle Flow by a Time-DependentMethod, " AIAA J., 12(4), Apr. 1974, p. 419.

9. Brown, E. F. and G. L. Hamilton, "A Survey of Methods for Exhaust-Nozzle Flow Analysis," AIAA Paper No. 60, 1975.

10. Hoglund, R. F., "Recent Advances in Gas-Particle Nozzle Flows,"ARS Journal, May 1962, p. 662.

11. Regan, J. F., H. D. Thompson, and R. F. Hoglund, "Tw:-DimensionalAnalysis of Transonic Gas-Particle Flows in Axisymmetric Nozzles, "J. Spacecraft, 8(4), Apr. 1971, p. 346.

12. Jacques, L.J. and J.A.M. Seguin, "Two-Dimensional Transonic Two-Phase Flow in Axisymmetric Nozzles, " AIAA Paper No. 74-1088,Oct. 1974.

-63-

REFERENCES (Continued)

13. Kliegel, J.R. and G.R. Nickerson, "Axisymmetric Two-PhasePerfect Gas Performance Program, " Report 02874-6006-ROOO, Vols. Iand II, Apr. 1967, TRW Systems Group, Redondo Beach, Ca. 90278.

14. Coats, D.E., et al., "A Computer Program for the Prediction of SolidPropellant Rocket Motor Performance, " Vols. I, I, and I1, AFRPL-TR-75-36, July 1975.

15. Soo, S. L., "Gas Dynamic Processes Involving Suspended Solids,"A.I. Ch.E. Journal, 7(3), Sept. 1961, p. 384.

16. Hultberg, J.A. and S. L. Soo, "Flow of a Gas-Solid Suspension Througha Nozzle, " AIAA Paper No. 65-6, Jan. 1965.

17. Thompson, J.F., F.C. Thames, and C.W. Martin, "Boundary-FittedCurvilinear Coordinates Systems for Solution of Partial DifferentialEquations on Fields Containing Any Number of Arbitrary Two-Dimen-sional Bodies, " NASA CR 2729, July 1977.

18. Henderson, C. B., "Drag Coefficients of Spheres in Continuum andRarefied Flows, " AIAA J., 14(6), June 1976, p. 707.

19. Carlson, D.J. and R.F. Hoglund, "Particle Drag and Heat Transfer inRocket Nozzles, " AIAA J., 2(11), 1964, p. 1980.

20. MacCormack, R. W., "The Effect of Viscosity in Hypervelocity ImpactCratering," AIAA Paper 69-354, May 1969.

21. Chang, I-Shih, "Three-Dimensional Supersonic Internal Flows,"AIAA Paper 76-423, July 1976.

22. Kutler, P., L. Sakell, and G. Aiello, "On the Shock-on-Shock InteractionProblem," AIAA Paper No. 74-524, June 1974.

23. Cuffel, R..F., L.H1-. Back, and P. F. Massier, "Transonic Flowfield in aSupersonic Nozzle with Small Throat Radius of Curvature, " AIAA J.,7(7), July 1969, p. 1364.

24. Chow, W. L. and I-Shih Chang, "Mach Reflection Associated with Over-expanded Nozzle Free Jet Flows, " AIAA J., 13(6), June 1975.

25. Chang, I-Shih and W. L. Chow, "Mach Disc from Underexpanded Axi-symmetric Nozzle Flow, " AIAA J., 12, Aug. 1974, p. 1079.

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REFERENCES (Concluded)

26. Private communication with Chemical Systems Division of UnitedTechnology Corporation on Titan III Engineering Drawings, Nov. 1978.

27. Private communication with R. Dunlap, Chemical Systems Division ofUnited Technology Corporation, Nov. 1978.

28. "Inertial Upper Stage SRM-II Baseline Design Review, " ChemicalSystems Division of United Technology Corporation, Dec. 1978.

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